Aside from 音読み (おんよみ) and 訓読み (くんよみ), there is a type of kanji reading called 当て字. This reading is the phonetic reading of a character, and there are three types.
1) The first type are words that use the phonetic reading of the characters. This is common in the kanji names of foreign countries. The names of countries were written using kanji that matched the phonetic pronunciations. The shortened versions today come from those original names.
America was written 亜米利加 (あまいりか) ➡ 米国 (べいこく) - rice
France was written 仏蘭西 (ふらんす) ➡ 仏国 (ふっこく) - Buddhism
Thus, the kanji associated with these two countries don’t necessarily relate to the countries themselves, but to the 当て字 used to write them. Most country names are now written in katakana.
America 亜米利加 (あまいりか) ➡ 米国 (べいこく) ➡ アメリカ
France 仏蘭西 (ふらんす) ➡ 仏国 (ふっこく)
Portugal 葡萄牙 (ぽるとがる) ➡ ポルトガル
寿司 (すし) is also an 当て字 word, which is why the word is made up of the characters for longevity and government official.
2) The second type are words that borrow the meaning of the characters but not the reading. These words derive from words that existed in Japanese before kanji were conscripted, and since there was no existing character for that word they borrowed a different kanji.
Tobacco (煙草 - たばこ) is made up the characters for smoke and grass, and the reading matches the loanword from the Portuguese, who brought it to Japan. Today it’s normally written in katakana.
Shrimp (海老 - えび) and seaweed (海苔 - のり) are also this type of 当て字.
3) The third type of 当て字 use both the reading and the meaning of the word. This comes from kanji that already matched the reading and meaning of words (usually foreign).
合羽 (かっぱ) is the one that you usually learn in Japanese class. This word means raincoat and was introduced by the Portuguese (capa). The kanji for join and wings were selected to make this word - apparently it reflects a bird’s wings coming together to protect from the rain - and the reading already matched.
Today, most foreign loanwords are written using katakana, so 当て字 words are usually from before the Meiji Era. Most country names and words like tobacco are written using katakana now, but you still see the kanji names of countries in the news.
当て字 is also commonly used in names. When parents select a name for a child, they want auspicious kanji that will match the reading but provide a meaning that will bring luck, make them a good person, or even have an auspicious number of strokes in the whole name.
Personal Note: In my husband’s family, the parents traditionally go to a fortune teller/priest to select a name that matches the future predictions for the child, will bring good fortune, and has an auspicious number of strokes including the last and first name. My husband and brother-in-law both had their names and kanji selected by the priest. My sister-in-law let the priest select her daughter’s name and kanji, but we wanted a particular name for our son and so we told him the name we had in mind and then he selected the kanji that best match the reading and last name. My husband, son, brother-in-law, and niece all have 当て字 in their names because of this selection process.